User blog:Rgetar/OCF based on ordinal array function
Yesterday I was thinking about my Veblen-like functions, and I realized that its weak point is that maximal number of iterations is ω, but ω is too small ordinal, and it makes rules more complex. Then I created OCF based on my ordinal array functions, since these functions do not have fixed points, and there should not be limited iterations. Also, today I slightly modified rules taken from ordinal array functions in order to make rules simpler, but, possibly, it makes calculations more difficult. Here I publish this OCF (currently I limited it by Ωω). Fundamental sequence and cofinality Apparently, there are different definitions of fundamental sequences and cofinality. I use the following definitions: *'Fundamental sequence' of ordinal α is increasing sequence of ordinals such as α is least ordinal greater than any element of this sequence. *'Cofinality' of ordinal α is number of elements of its shortest possible fundamental sequence. So, cofinality of successor ordinal α + 1 is 1, and its shortest possible fundamental sequence contains only one element (α + 1)0 = α Cofinality of 0 is 0, so, 0 does not have fundamental sequences. Rules Here n is natural number, α, γ, X are ordinals, and card(α) is cardinality of ordinal α. Ordinals are defined using fundamental sequences, so, this is also fundamental sequence system. To find ordinal α, we need cofinality of α and function from an ordinal to element of fundamental sequence of α. Rules: *0 has not fundamental sequences *(α + 1)0 = α *Ωnγ = γ *0α = α + 1 (that is 0α0 = α) *+ 1α = XXα *(Xα)γ = [Xγ]α, if ω ≤ cof(X) ≤ card(α) *Xα = [Xα]α, if cof(X) > card(α) (Note: rules 5 and 7 do not define fundamental sequences directly, but using these rules it can be reduced to other rules). Cofinalities: *cof(0) = 0 *cof(α + 1) = 1 *cof(Ωn) = Ωn *cof(0α) = 1 *cof(+ 1α) = cof(XXα) *cof(Xα) = cof(X), if ω ≤ cof(X) ≤ card(α) *cof(Xα) = cof([Xα]α), if cof(X) > card(α) Standard form n, Ωn are in standard form. For the rest of ordinals standard form is X0X1X2X3X4X5...Ωn with any integer positive number of "Xi"`s, where X0 < X1 < X2 < X3 < X4 < X5 < ... and Xα does not contain "α". Currently I do not know, how to convert non-standard form of ordinals written this way to standard form, and how to compare them. But, I think, this may be not so important, since when we find fundamental sequence elements for standard forms, we always (unlike Veblen functions) get ordinals also in standard forms. Examples We may start from ω and use the following rules: *Go to successor of current ordinal α (that is α → 0α = α + 1) *If we get XXα, then replace XX with [0X] (that is XXα → [0X]α) *If we get Xα, where X contains α, and card(α) = Ωn, then replace α with + 1 (that is α → + 1) So, we are starting from ω (standard forms of ordinals we will make blue): *ω Now use α → 0α rule: *0ω = ω + 1 Now use α → 0α rule again: *00ω = ω + 2 But we got XXα, where X = 0, and α = ω. So, use XXα → [0X]α rule: *00ω = [00]ω = ω + 2 Since 00 = 0 + 1 = 1, and we should write natural numbers directly, not as expressions, then *[00]ω = 1ω = ω + 2 Use α → 0α rule again: *01ω = ω + 3 And again: *001ω = ω + 4 We got XXα, now X = 0, and α = 1ω = ω + 2. Use XXα → [0X]α rule: *001ω = [00]1ω = 11ω = ω + 4 Use XXα → [0X]α rule one more time: *11ω = [01]ω = ω + 4 Since 01 = 1 + 1 = 2 then *[01]ω = 2ω = ω + 4 Continue: *02ω = ω + 5 *002ω = 12ω = ω + 6 *012ω = ω + 7 *0012ω = 112ω = 22ω = 3ω = ω + 8 *03ω = ω + 9 *003ω = 13ω = ω + 10 *013ω = ω + 11 *0013ω = 113ω = 23ω = ω + 12 *023ω = ω + 13 *0023ω = 123ω = ω + 14 *0123ω = ω + 15 *00123ω = 1123ω = 223ω = 33ω = 4ω = ω + 16 *04ω = ω + 17 *004ω = 14ω = ω + 18 *014ω = ω + 19 *0014ω = 114ω = 24ω = ω + 20 *024ω = ω + 21 *0024ω = 124ω = ω + 22 *0124ω = ω + 23 *00124ω = 1124ω = 224ω = 34ω = ω + 24 *034ω = ω + 25 *0034ω = 134ω = ω + 26 *0134ω = ω + 27 *00134ω = 1134ω = 234ω = ω + 28 *0234ω = ω + 29 *00234ω = 1234ω = ω + 30 *01234ω = ω + 31 *001234ω = 11234ω = 2234ω = 334ω = 44ω = 5ω = ω + 32 *... *6ω = ω + 64 *7ω = ω + 128 *8ω = ω + 256 *9ω = ω + 512 *10ω = ω + 1024 *... Now we go to limit: *ωω = ω + ω = ω2 We got Xα with "α", where α = ω, and Ωn = card(ω) = Ω0 = ω, so n = 0. So, use the third rule α → + 1, where Ωn + 1 = Ω1 = Ω, that is replace "ω" with "Ω": *ωω = Ωω = ω2 Continue: *0Ωω = ω2 + 1 *00Ωω = 1Ωω = ω2 + 2 *01Ωω = ω2 + 3 *001Ωω = 11Ωω = 2Ωω = ω2 + 4 *02Ωω = ω2 + 5 *002Ωω = 12Ωω = ω2 + 6 *012Ωω = ω2 + 7 *0012Ωω = 112Ωω = 22Ωω = 3Ωω = ω2 + 8 *... *4Ωω = ω2 + 16 *5Ωω = ω2 + 32 *6Ωω = ω2 + 64 *7Ωω = ω2 + 128 *8Ωω = ω2 + 256 *... *ωΩω = ω2 + ω = ω3 Note: this is not the case Xα with "α", because here X = ω, and α = Ωω, so, we do not use rule α → + 1. Continue: *0ωΩω = ω3 + 1 *00ωΩω = 1ωΩω = ω3 + 2 *01ωΩω = ω3 + 3 *001ωΩω = 11ωΩω = 2ωΩω = ω3 + 4 *02ωΩω = ω3 + 5 *002ωΩω = 12ωΩω = ω3 + 6 *012ωΩω = ω3 + 7 *0012ωΩω = 112ωΩω = 22ωΩω = 3ωΩω = ω3 + 8 *... *4ωΩω = ω3 + 16 *5ωΩω = ω3 + 32 *6ωΩω = ω3 + 64 *7ωΩω = ω3 + 128 *8ωΩω = ω3 + 256 *... *ωωΩω = ω3 + ω = ω4 We got XXα, where X = ω, and α = Ωω = ω2. So, use XXα → [0X]α rule: *ωωΩω = [0ω]Ωω = + 1Ωω = ω4 Continue: *0[0ω]Ωω = 0+ 1Ωω = ω4 + 1 *00[0ω]Ωω = 1[0ω]Ωω = 1+ 1Ωω = ω4 + 2 *01[0ω]Ωω = 01+ 1Ωω = ω4 + 3 *001[0ω]Ωω = 11[0ω]Ωω = 2[0ω]Ωω = 2+ 1Ωω = ω4 + 4 *02[0ω]Ωω = 02+ 1Ωω = ω4 + 5 *002[0ω]Ωω = 12[0ω]Ωω = 12+ 1Ωω = ω4 + 6 *012[0ω]Ωω = 012+ 1Ωω = ω4 + 7 *0012[0ω]Ωω = 112[0ω]Ωω = 22[0ω]Ωω = 3[0ω]Ωω = 3+ 1Ωω = ω4 + 8 *... *4[0ω]Ωω = 4+ 1Ωω = ω4 + 16 *5[0ω]Ωω = 5+ 1Ωω = ω4 + 32 *6[0ω]Ωω = 6+ 1Ωω = ω4 + 64 *7[0ω]Ωω = 7+ 1Ωω = ω4 + 128 *8[0ω]Ωω = 8+ 1Ωω = ω4 + 256 *... *ω[0ω]Ωω = ω+ 1Ωω = ω4 + ω = ω5 *0ω[0ω]Ωω = 0ω+ 1Ωω = ω5 + 1 *1ω[0ω]Ωω = 1ω+ 1Ωω = ω5 + 2 *2ω[0ω]Ωω = 2ω+ 1Ωω = ω5 + 4 *3ω[0ω]Ωω = 3ω+ 1Ωω = ω5 + 8 *4ω[0ω]Ωω = 4ω+ 1Ωω = ω5 + 16 *5ω[0ω]Ωω = 5ω+ 1Ωω = ω5 + 32 *... Triple using of rule XXα → [0X]α: *ωω[0ω]Ωω = [0ω][0ω]Ωω = [00ω]Ωω = [[00]ω]Ωω = [1ω]Ωω = + 2Ωω = ω5 + ω = ω6 *0[1ω]Ωω = 0+ 2Ωω = ω6 + 1 *1[1ω]Ωω = 1+ 2Ωω = ω6 + 2 *2[1ω]Ωω = 2+ 2Ωω = ω6 + 4 *3[1ω]Ωω = 3+ 2Ωω = ω6 + 8 *4[1ω]Ωω = 4+ 2Ωω = ω6 + 16 *5[1ω]Ωω = 5+ 2Ωω = ω6 + 32 *... *ω[1ω]Ωω = ω+ 2Ωω = ω7 *ωω[1ω]Ωω = [0ω][1ω]Ωω = + 1+ 2Ωω = ω8 *ω[0ω][1ω]Ωω = ω+ 1+ 2Ωω = ω9 *ωω[0ω][1ω]Ωω = [0ω][0ω][1ω]Ωω = [00ω][1ω]Ωω = [1ω][1ω]Ωω = [01ω]Ωω = + 3Ωω = ω10 *ω[01ω]Ωω = ω+ 3Ωω = ω11 *ωω[01ω]Ωω = [0ω][01ω]Ωω = + 1+ 3Ωω = ω12 *ω[0ω][01ω]Ωω = ω+ 1+ 3Ωω = ω13 *ωω[0ω][01ω]Ωω = [0ω][0ω][01ω]Ωω = [1ω][01ω]Ωω = + 2+ 3Ωω = ω14 *ω[1ω][01ω]Ωω = ω+ 2+ 3Ωω = ω15 *ωω[1ω][01ω]Ωω = [0ω][1ω][01ω]Ωω = + 1+ 2+ 3Ωω = ω16 *ω[0ω][1ω][01ω]Ωω = ω+ 1+ 2+ 3Ωω = ω17 *ωω[0ω][1ω][01ω]Ωω = [0ω][0ω][1ω][01ω]Ωω = [1ω][1ω][01ω]Ωω = [01ω][01ω]Ωω = [001ω]Ωω = [11ω]Ωω = [2ω]Ωω = + 4Ωω = ω18 *... *[02ω]Ωω = + 5Ωω = ω34 *[12ω]Ωω = + 6Ωω = ω66 *[012ω]Ωω = + 7Ωω = ω130 *[3ω]Ωω = + 8Ωω = ω258 *... *[4ω]Ωω = + 16Ωω = ω65538 *[5ω]Ωω = + 32Ωω = ω4294967298 *[6ω]Ωω = + 64Ωω = ω18446744073709551618 *[7ω]Ωω = + 128Ωω = ω340282366920938463463374607431768211458 *[8ω]Ωω = + 256Ωω = ω115792089237316195423570985008687907853269984665640564039457584007913129639938 *... *[ωω]Ωω = + ωΩω = ω2Ωω = ω2 We got Xα with α, where X = α = ω, card(α) = Ωn = ω, n = 0. Use rule α → + 1, that is ω → Ω: *[ωω]Ωω = [Ωω]Ωω = ω2 We got Xα with α, where X = α = Ωω, card(α) = Ωn = ω, n = 0. Use rule α → + 1, that is [Ωω] → Ω: *[Ωω]Ωω = ΩΩω = ω2 Finally, use rule XXα → [0X]α: *ΩΩω = [0Ω]ω = + 1ω = ω2 Continue: *0[0Ω]ω = 0+ 1ω = ω2 + 1 *1[0Ω]ω = 1+ 1ω = ω2 + 2 *2[0Ω]ω = 2+ 1ω = ω2 + 3 *3[0Ω]ω = 3+ 1ω = ω2 + 4 *4[0Ω]ω = 4+ 1ω = ω2 + 5 *5[0Ω]ω = 5+ 1ω = ω2 + 6 *... *ω[0Ω]ω = ω+ 1ω = ω2 + ω *[0ω][0Ω]ω = + 1+ 1ω = ω2 + ω2 *ω[0ω][0Ω]ω = ω+ 1+ 1ω = ω2 + ω3 *[1ω][0Ω]ω = + 2+ 1ω = ω2 + ω4 *ω[1ω][0Ω]ω = ω+ 2+ 1ω = ω2 + ω5 *[0ω][1ω][0Ω]ω = + 1+ 2+ 1ω = ω2 + ω6 *ω[0ω][1ω][0Ω]ω = ω+ 1+ 2+ 1ω = ω2 + ω7 *[01ω][0Ω]ω = + 3+ 1ω = ω2 + ω8 *... *[2ω][0Ω]ω = + 4+ 1ω = ω2 + ω16 *[02ω][0Ω]ω = + 5+ 1ω = ω2 + ω32 *[12ω][0Ω]ω = + 6+ 1ω = ω2 + ω64 *[012ω][0Ω]ω = + 7+ 1ω = ω2 + ω128 *[3ω][0Ω]ω = + 8+ 1ω = ω2 + ω128 *... *[4ω][0Ω]ω = + 16+ 1ω = ω2 + ω65536 *[5ω][0Ω]ω = + 32+ 1ω = ω2 + ω4294967296 *[6ω][0Ω]ω = + 64+ 1ω = ω2 + ω18446744073709551616 *[7ω][0Ω]ω = + 128+ 1ω = ω2 + ω340282366920938463463374607431768211456 *[8ω][0Ω]ω = + 256+ 1ω = ω2 + ω115792089237316195423570985008687907853269984665640564039457584007913129639936 *... *[ωω][0Ω]ω = [Ωω][0Ω]ω = ω2+ 1ω = ω22 *0[Ωω][0Ω]ω = ω2+ 1ω + 1 = ω22 + 1 *1[Ωω][0Ω]ω = ω2+ 1ω + 2 = ω22 + 2 *2[Ωω][0Ω]ω = ω2+ 1ω + 3 = ω22 + 3 *3[Ωω][0Ω]ω = ω2+ 1ω + 4 = ω22 + 4 *... *ω[Ωω][0Ω]ω = ωω2+ 1ω = ω22 + ω *[0ω][Ωω][0Ω]ω = + 1ω2+ 1ω = ω22 + ω2 *[1ω][Ωω][0Ω]ω = + 2ω2+ 1ω = ω22 + ω4 *[2ω][Ωω][0Ω]ω = + 4ω2+ 1ω = ω22 + ω16 *[3ω][Ωω][0Ω]ω = + 8ω2+ 1ω = ω22 + ω256 *[4ω][Ωω][0Ω]ω = + 16ω2+ 1ω = ω22 + ω65536 *[5ω][Ωω][0Ω]ω = + 32ω2+ 1ω = ω22 + ω4294967296 *... *[ωω][Ωω][0Ω]ω = [Ωω][Ωω][0Ω]ω = [0Ωω][0Ω]ω = + 1+ 1ω = ω23 *ω[0Ωω][0Ω]ω = ω+ 1+ 1ω = ω23 + ω *[0ω][0Ωω][0Ω]ω = + 1+ 1+ 1ω = ω23 + ω2 *[1ω][0Ωω][0Ω]ω = + 2+ 1+ 1ω = ω23 + ω4 *[2ω][0Ωω][0Ω]ω = + 4+ 1+ 1ω = ω23 + ω16 *[3ω][0Ωω][0Ω]ω = + 8+ 1+ 1ω = ω23 + ω256 *[4ω][0Ωω][0Ω]ω = + 16+ 1+ 1ω = ω23 + ω65536 *[5ω][0Ωω][0Ω]ω = + 32+ 1+ 1ω = ω23 + ω4294967296 *... *[ωω][0Ωω][0Ω]ω = [Ωω][0Ωω][0Ω]ω = ω2+ 1+ 1ω = ω24 *[Ωω][Ωω][0Ωω][0Ω]ω = [0Ωω][0Ωω][0Ω]ω = [00Ωω][0Ω]ω = [1Ωω][0Ω]ω = + 2+ 1ω = ω25 *[Ωω][1Ωω][0Ω]ω = ω2+ 2+ 1ω = ω26 *[Ωω][Ωω][1Ωω][0Ω]ω = [0Ωω][1Ωω][0Ω]ω = + 1+ 2+ 1ω = ω27 *[Ωω][0Ωω][1Ωω][0Ω]ω = ω2+ 1+ 2+ 1ω = ω28 *[Ωω][Ωω][0Ωω][1Ωω][0Ω]ω = [0Ωω][0Ωω][1Ωω][0Ω]ω = [00Ωω][1Ωω][0Ω]ω = [1Ωω][1Ωω][0Ω]ω = [01Ωω][0Ω]ω = + 3+ 1ω = ω29 *... *[2Ωω][0Ω]ω = + 4+ 1ω = ω217 *[02Ωω][0Ω]ω = + 5+ 1ω = ω233 *[12Ωω][0Ω]ω = + 6+ 1ω = ω265 *[012Ωω][0Ω]ω = + 7+ 1ω = ω2129 *[3Ωω][0Ω]ω = + 8+ 1ω = ω2257 *... *[4Ωω][0Ω]ω = + 16+ 1ω = ω265537 *[5Ωω][0Ω]ω = + 32+ 1ω = ω24294967297 *[6Ωω][0Ω]ω = + 64+ 1ω = ω218446744073709551617 *[7Ωω][0Ω]ω = + 128+ 1ω = ω2340282366920938463463374607431768211457 *[8Ωω][0Ω]ω = + 256+ 1ω = ω2115792089237316195423570985008687907853269984665640564039457584007913129639937 *... *[ωΩω][0Ω]ω = ω3+ 1ω = ω3 *[ωΩω][ωΩω][0Ω]ω = [0ωΩω][0Ω]ω = + 1+ 1ω = ω32 *[ωΩω][0ωΩω][0Ω]ω = ω3+ 1+ 1ω = ω33 *[ωΩω][ωΩω][0ωΩω][0Ω]ω = [0ωΩω][0ωΩω][0Ω]ω = [1ωΩω][0Ω]ω = + 2+ 1ω = ω34 *... *[2ωΩω][0Ω]ω = + 4+ 1ω = ω316 *[3ωΩω][0Ω]ω = + 8+ 1ω = ω3256 *[4ωΩω][0Ω]ω = + 16+ 1ω = ω365536 *[5ωΩω][0Ω]ω = + 32+ 1ω = ω34294967296 *... *[ωωΩω][0Ω]ω = [[0ω]Ωω][0Ω]ω = ω4+ 1ω = ω4 *[0[0ω]Ωω][0Ω]ω = + 1+ 1ω = ω42 *[1[0ω]Ωω][0Ω]ω = + 2+ 1ω = ω44 *[2[0ω]Ωω][0Ω]ω = + 4+ 1ω = ω416 *[3[0ω]Ωω][0Ω]ω = + 8+ 1ω = ω4256 *[4[0ω]Ωω][0Ω]ω = + 16+ 1ω = ω465536 *[5[0ω]Ωω][0Ω]ω = + 32+ 1ω = ω34294967296 *... *[ω[0ω]Ωω][0Ω]ω = ω5+ 1ω = ω5 *[[1ω]Ωω][0Ω]ω = ω6+ 1ω = ω6 *[ω[1ω]Ωω][0Ω]ω = ω7+ 1ω = ω7 *[[0ω][1ω]Ωω][0Ω]ω = ω8+ 1ω = ω8 *[ω[0ω][1ω]Ωω][0Ω]ω = ω9+ 1ω = ω9 *[[01ω]Ωω][0Ω]ω = ω10+ 1ω = ω10 *[ω[01ω]Ωω][0Ω]ω = ω11+ 1ω = ω11 *[[0ω][01ω]Ωω][0Ω]ω = ω12+ 1ω = ω12 *[ω[0ω][01ω]Ωω][0Ω]ω = ω13+ 1ω = ω13 *[[1ω][01ω]Ωω][0Ω]ω = ω14+ 1ω = ω14 *[ω[1ω][01ω]Ωω][0Ω]ω = ω15+ 1ω = ω15 *[[0ω][1ω][01ω]Ωω][0Ω]ω = ω16+ 1ω = ω16 *[ω[0ω][1ω][01ω]Ωω][0Ω]ω = ω17+ 1ω = ω17 *[[2ω]Ωω][0Ω]ω = ω18+ 1ω = ω18 *... *[[02ω]Ωω][0Ω]ω = ω34+ 1ω = ω34 *[[12ω]Ωω][0Ω]ω = ω66+ 1ω = ω66 *[[012ω]Ωω][0Ω]ω = ω130+ 1ω = ω130 *[[3ω]Ωω][0Ω]ω = ω258+ 1ω = ω258 *... *[[4ω]Ωω][0Ω]ω = ω65538+ 1ω = ω65538 *[[5ω]Ωω][0Ω]ω = ω4294967298+ 1ω = ω4294967298 *[[6ω]Ωω][0Ω]ω = ω18446744073709551618+ 1ω = ω18446744073709551618 *[[7ω]Ωω][0Ω]ω = ω340282366920938463463374607431768211458+ 1ω = ω340282366920938463463374607431768211458 *[[8ω]Ωω][0Ω]ω = ω115792089237316195423570985008687907853269984665640564039457584007913129639938+ 1ω = ω115792089237316195423570985008687907853269984665640564039457584007913129639938 *... *[[ωω]Ωω][0Ω]ω = [[Ωω]Ωω][0Ω]ω = [ΩΩω][0Ω]ω = [[0Ω]ω][0Ω]ω = Ω[0Ω]ω = Ω+ 1ω = ωω *[[0Ω]ω]Ω[0Ω]ω = ω2Ω+ 1ω = ωω2 *[[0Ω]ω][[0Ω]ω]Ω[0Ω]ω = [0[0Ω]ω]Ω[0Ω]ω = + 1Ω+ 1ω = ωω3 *[[0Ω]ω][0[0Ω]ω]Ω[0Ω]ω = ω2+ 1Ω+ 1ω = ωω4 *[[0Ω]ω][[0Ω]ω][0[0Ω]ω]Ω[0Ω]ω = [0[0Ω]ω][0[0Ω]ω]Ω[0Ω]ω = [1[0Ω]ω]Ω[0Ω]ω = + 2Ω+ 1ω = ωω5 *... *[01[0Ω]ω]Ω[0Ω]ω = + 3Ω+ 1ω = ωω9 *[2[0Ω]ω]Ω[0Ω]ω = + 4Ω+ 1ω = ωω17 *[02[0Ω]ω]Ω[0Ω]ω = + 5Ω+ 1ω = ωω33 *[12[0Ω]ω]Ω[0Ω]ω = + 6Ω+ 1ω = ωω65 *[012[0Ω]ω]Ω[0Ω]ω = + 7Ω+ 1ω = ωω129 *[3[0Ω]ω]Ω[0Ω]ω = + 8Ω+ 1ω = ωω257 *... *[4[0Ω]ω]Ω[0Ω]ω = + 16Ω+ 1ω = ωω65537 *[5[0Ω]ω]Ω[0Ω]ω = + 32Ω+ 1ω = ωω4294967297 *[6[0Ω]ω]Ω[0Ω]ω = + 64Ω+ 1ω = ωω18446744073709551617 *... *[ω[0Ω]ω]Ω[0Ω]ω = + ωΩ+ 1ω = ωω + 1 *[[0ω][0Ω]ω]Ω[0Ω]ω = + ω2Ω+ 1ω = ωω + 2 *[ω[0ω][0Ω]ω]Ω[0Ω]ω = + ω3Ω+ 1ω = ωω + 3 *[[1ω][0Ω]ω]Ω[0Ω]ω = + ω4Ω+ 1ω = ωω + 4 *[ω[1ω][0Ω]ω]Ω[0Ω]ω = + ω5Ω+ 1ω = ωω + 5 *[[0ω][1ω][0Ω]ω]Ω[0Ω]ω = + ω6Ω+ 1ω = ωω + 6 *[ω[0ω][1ω][0Ω]ω]Ω[0Ω]ω = + ω7Ω+ 1ω = ωω + 7 *[[01ω][0Ω]ω]Ω[0Ω]ω = + ω8Ω+ 1ω = ωω + 8 *... *[[ωω][0Ω]ω]Ω[0Ω]ω = [[Ωω][0Ω]ω]Ω[0Ω]ω = ω22Ω+ 1ω = ωω2 *[[0Ωω][0Ω]ω]Ω[0Ω]ω = ω23Ω+ 1ω = ωω3 *[[Ωω][0Ωω][0Ω]ω]Ω[0Ω]ω = ω24Ω+ 1ω = ωω4 *[[1Ωω][0Ω]ω]Ω[0Ω]ω = ω25Ω+ 1ω = ωω5 *... *[[ωΩω][0Ω]ω]Ω[0Ω]ω = ω3Ω+ 1ω = ωω2 *[[[0ω]Ωω][0Ω]ω]Ω[0Ω]ω = ω4Ω+ 1ω = ωω3 *[[ω[0ω]Ωω][0Ω]ω]Ω[0Ω]ω = ω5Ω+ 1ω = ωω4 *[[[1ω]Ωω][0Ω]ω]Ω[0Ω]ω = ω6Ω+ 1ω = ωω5 *... *[[[ωω]Ωω][0Ω]ω]Ω[0Ω]ω = [[[Ωω]Ωω][0Ω]ω]Ω[0Ω]ω = [[ΩΩω][0Ω]ω]Ω[0Ω]ω = [[[0Ω]ω][0Ω]ω]Ω[0Ω]ω = [Ω[0Ω]ω]Ω[0Ω]ω = ΩΩ[0Ω]ω = [0Ω][0Ω]ω = [00Ω]ω = [1Ω]ω = + 2ω = ωωω *Ω[1Ω]ω = Ω+ 2ω = ωωωω *[0Ω][1Ω]ω = + 1+ 2ω = ωωωωω *Ω[0Ω][1Ω]ω = Ω+ 1+ 2ω = ωωωωωω *[01Ω]ω = + 3ω = ωωωωωωω *... *[2Ω]ω = + 4ω = ωωωωωωωωωωωωωωω *[02Ω]ω = + 5ω = ωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωω *[12Ω]ω = + 6ω = ωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωωω *... *[ωΩ]ω = ε0 We got Xα with α, where α = ω, X = ωΩ, Ωn = card(ω) = ω, n = 0. Use rule α → + 1, that is ω → Ω: *[ωΩ]ω = [ΩΩ]ω = ε0 We got Xα with α, where α = X = Ω, Ωn = card(Ω) = Ω1 = Ω, n = 1. Use rule α → + 1, that is Ω → Ω2: *[ΩΩ]ω = [Ω2Ω]ω = Ω2ω = ε0 Continue: *0[Ω2Ω]ω = 0Ω2ω = ε0 + 1 *1[Ω2Ω]ω = 1Ω2ω = ε0 + 2 *01[Ω2Ω]ω = 01Ω2ω = ε0 + 3 *2[Ω2Ω]ω = 2Ω2ω = ε0 + 4 *... *ω[Ω2Ω]ω = ωΩ2ω = ε0 + ω *[0ω][Ω2Ω]ω = + 1Ω2ω = ε0 + ω2 *ω[0ω][Ω2Ω]ω = ω+ 1Ω2ω = ε0 + ω3 *... *[Ωω][Ω2Ω]ω = ω2Ω2ω = ε0 + ω2 *[ωΩω][Ω2Ω]ω = ω3Ω2ω = ε0 + ω3 *[[0ω]Ωω][Ω2Ω]ω = ω4Ω2ω = ε0 + ω4 *[ω[0ω]Ωω][Ω2Ω]ω = ω5Ω2ω = ε0 + ω5 *... *[[0Ω]ω][Ω2Ω]ω = ω2Ω2ω = ε0 + ωω *[Ω[0Ω]ω][Ω2Ω]ω = ωωΩ2ω = ε0 + ωωω *[[1Ω]ω][Ω2Ω]ω = ωωωΩ2ω = ε0 + ωωωω *[Ω[1Ω]ω][Ω2Ω]ω = ωωωωΩ2ω = ε0 + ωωωωω *... *[[ωΩ]ω][Ω2Ω]ω = Ω[Ω2Ω]ω = ε0Ω2ω = ε02 *[[Ω2Ω]ω]Ω[Ω2Ω]ω = ε0ΩΩ2ω = ε03 *[0[Ω2Ω]ω]Ω[Ω2Ω]ω = + 1ΩΩ2ω = ε04 *[[Ω2Ω]ω][0[Ω2Ω]ω]Ω[Ω2Ω]ω = ε0+ 1ΩΩ2ω = ε05 *... *[ω[Ω2Ω]ω]Ω[Ω2Ω]ω = + ωΩΩ2ω = ωε0 + 1 *[[0ω][Ω2Ω]ω]Ω[Ω2Ω]ω = + ω2ΩΩ2ω = ωε0 + 2 *[ω[0ω][Ω2Ω]ω]Ω[Ω2Ω]ω = + ω3ΩΩ2ω = ωε0 + 3 *... *[[Ωω][Ω2Ω]ω]Ω[Ω2Ω]ω = + ω2ΩΩ2ω = ωε0 + ω *[[[0Ω]ω][Ω2Ω]ω]Ω[Ω2Ω]ω = + ωωΩΩ2ω = ωε0 + ωω *[[Ω[0Ω]ω][Ω2Ω]ω]Ω[Ω2Ω]ω = + ωωωΩΩ2ω = ωε0 + ωωω *[[[1Ω]ω][Ω2Ω]ω]Ω[Ω2Ω]ω = + ωωωωΩΩ2ω = ωε0 + ωωωω *[[Ω[1Ω]ω][Ω2Ω]ω]Ω[Ω2Ω]ω = + ωωωωωΩΩ2ω = ωε0 + ωωωωω *... *[0Ω][Ω2Ω]ω = + 1Ω2ω = ωε02 *Ω[0Ω][Ω2Ω]ω = Ω+ 1Ω2ω = ωωε02 *1Ω][Ω2Ω]ω = + 2Ω2ω = ωωωε02 *Ω1Ω][Ω2Ω]ω = Ω+ 2Ω2ω = ωωωωε02 *[0Ω]1Ω][Ω2Ω]ω = + 1+ 2Ω2ω = ωωωωωε02 *Ω[0Ω]1Ω][Ω2Ω]ω = Ω+ 1+ 2Ω2ω = ωωωωωωε02 *[01Ω][Ω2Ω]ω = + 3Ω2ω = ωωωωωωωε02 *... *[ωΩ][Ω2Ω]ω = + ωΩ2ω = ε1 *Ω[ωΩ][Ω2Ω]ω = Ω+ ωΩ2ω = ε12 *[0Ω][ωΩ][Ω2Ω]ω = + 1+ ωΩ2ω = ωε12 *Ω[0Ω][ωΩ][Ω2Ω]ω = Ω+ 1+ ωΩ2ω = ωωε12 *[1Ω][ωΩ][Ω2Ω]ω = + 2+ ωΩ2ω = ωωωε12 *... *[0ωΩ][Ω2Ω]ω = + ω + 1Ω2ω = ε2 *[ωΩ][0ωΩ][Ω2Ω]ω = + ω+ ω + 1Ω2ω = ε3 *[1ωΩ][Ω2Ω]ω = + ω + 2Ω2ω = ε4 *[ωΩ][1ωΩ][Ω2Ω]ω = + ω+ ω + 2Ω2ω = ε5 *[0ωΩ][1ωΩ][Ω2Ω]ω = + ω + 1+ ω + 2Ω2ω = ε6 *[ωΩ][0ωΩ][1ωΩ][Ω2Ω]ω = + ω+ ω + 1+ ω + 2Ω2ω = ε7 *[01ωΩ][Ω2Ω]ω = + ω + 3Ω2ω = ε8 *... *[[0ω]Ω][Ω2Ω]ω = + ω2Ω2ω = εω *[ωΩ][[0ω]Ω][Ω2Ω]ω = + ω+ ω2Ω2ω = εω + 1 *[0ωΩ][[0ω]Ω][Ω2Ω]ω = + ω + 1+ ω2Ω2ω = εω + 2 *[ωΩ][0ωΩ][[0ω]Ω][Ω2Ω]ω = + ω+ ω + 1+ ω2Ω2ω = εω + 3 *[1ωΩ][[0ω]Ω][Ω2Ω]ω = + ω + 2+ ω2Ω2ω = εω + 4 *... *[0[0ω]Ω][Ω2Ω]ω = + ω2 + 1Ω2ω = εω2 *[[0ω]Ω][0[0ω]Ω][Ω2Ω]ω = + ω2+ ω2 + 1Ω2ω = εω3 *[1[0ω]Ω][Ω2Ω]ω = + ω2 + 2Ω2ω = εω4 *... *[ω[0ω]Ω][Ω2Ω]ω = + ω3Ω2ω = εω2 *[0ω[0ω]Ω][Ω2Ω]ω = + ω3 + 1Ω2ω = εω22 *[ω[0ω]Ω][0ω[0ω]Ω][Ω2Ω]ω = + ω3+ ω3 + 1Ω2ω = εω23 *[1ω[0ω]Ω][Ω2Ω]ω = + ω3 + 2Ω2ω = εω24 *... *[[1ω]Ω][Ω2Ω]ω = + ω4Ω2ω = εω3 *[ω[1ω]Ω][Ω2Ω]ω = + ω5Ω2ω = εω4 *[[0ω][1ω]Ω][Ω2Ω]ω = + ω6Ω2ω = εω5 *[ω[0ω][1ω]Ω][Ω2Ω]ω = + ω7Ω2ω = εω6 *[[2ω]Ω][Ω2Ω]ω = + ω8Ω2ω = εω7 *... *[[Ωω]Ω][Ω2Ω]ω = + ω2Ω2ω = εωω (wip) Category:Blog posts